On 12 Apr, 08:21, fom <fomJ...@nyms.net> wrote: > On 4/12/2013 12:57 AM, WM wrote: > > > Matheology § 246 > > > Cantor's list contains real numbers r as binary or decimal fractions. > > WM is wrong again. > > Cantor's list consists of one representation for > a real number that is not on the given list purported > to consist of representations for every number. > > > Real numbers, however, are /limits/ of binary or decimal fractions. > > Yes. This is why the arithmetic of real numbers is not the > arithmetic of rational numbers, although the latter is > representable within the former. > > > For every terminating fraction of r, Cantor obtains a difference > > between r and the due terminating fraction of the anti-diagonal d: > > r_nn =/= d_n. > > He does not obtain one. He constructs one based on > syntactic criteria. > > > He concludes that this remains true for the limits of > > the list numbers r and d by using the argument: different sequences > > have different limits. But it is well known that this argument is not > > admissible in proofs because it is false. > > But, the argument is based on the representation > of real numbers with respect to representation > according to the output of the Euclidean algorithm. > > There is no assumption concerning the convergence > of partial sums whatsoever. > > If WM's statement were to be given credence, the Euclidean > algorithm of long division could no longer be considered > as providing a faithful representation of distinct real > numbers (or rational numbers for that matter).
0.333... and so on is not 1/3 in any digit place, only in the imagination of infinite actual representation, you can do your long division in infinity that number series will *never* represent 1/3. Partition of the reals into bases using longdivision is not lossless to use computer terms. It is 0.333... is an identity now they also claim 1=1.000... is an identity and it is laughable there is no such creature. The natural 1 is discete is does not have any decimal expansion. The zeros is dreamed up from some noneexistent numberline.